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Polynomial Differential Form
In algebra, the ring of polynomial differential forms on the standard ''n''-simplex is the differential graded algebra: :\Omega^*_( = \mathbb _0, ..., t_n, dt_0, ..., dt_n(\sum t_i - 1, \sum dt_i). Varying ''n'', it determines the simplicial commutative dg algebra: :\Omega^*_ (each u: \to /math> induces the map \Omega^*_( \to \Omega^*_( , t_i \mapsto \sum_ t_j). References * Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL De Rham Theory and Rational Homotopy Type In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ..., Memoirs of the A. M. S., vol. 179, 1976. * External links * https://ncatlab.org/nlab/show/differential+forms+on+simplices * https://mathoverflow.net/questions/220532/polynomial-differential-forms-on-bg Differential algebra Ring theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular polytope. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Graded Algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. __TOC__ Definition A differential graded algebra (or DG-algebra for short) ''A'' is a graded algebra equipped with a map d\colon A \to A which has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions: A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the monoidal category of chain complexes. A DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential ''d''. A differential graded augmented algebra (also called a DGA-algebra, an augmented DG-algebra or simply a DGA) is a DG-algebra equipped with a DG morphism to the ground ring (the terminology is due to Henri Cartan). ''Warning:'' some sources use the term ''DGA'' for a DG-alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Commutative Ring
In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplicial commutative ring, then it can be shown that \pi_0 A is a ring and \pi_i A are modules over that ring (in fact, \pi_* A is a graded ring over \pi_0 A.) A topology-counterpart of this notion is a commutative ring spectrum. Examples *The ring of polynomial differential forms on simplexes. Graded ring structure Let ''A'' be a simplicial commutative ring. Then the ring structure of ''A'' gives \pi_* A = \oplus_ \pi_i A the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, \pi_* A is the homology of the chain complex corresponding to ''A''; in particular, it is a graded abelian group. Next, to multiply two elements, writing S^1 for the simplicial circle, let x:(S^1)^ \to A, \, \, y:(S^1)^ \to A be two maps. Then the composition :(S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Theorem
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \to \cdo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Homotopy Type
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homotopy theory makes certain calculations much easier. Rational homotopy types of simply connected spaces can be identified with (isomorphism classes of) certain algebraic objects called Sullivan minimal models, which are commutative differential graded algebras over the rational numbers satisfying certain conditions. A geometric application was the theorem of Sullivan and Micheline Vigué-Poirrier (1976): every simply connected closed Riemannian manifold ''X'' whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct closed geodesics. The proof used rational homotopy theory to show that the Betti numbers of the free loop space of ''X'' are unbounded. The theorem then follows from a 1969 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natural example of a differential field is the field of rational functions in one variable over the complex numbers, \mathbb(t), where the derivation is differentiation with respect to t. Differential algebra refers also to the area of mathematics consisting in the study of these algebraic objects and their use in the algebraic study of differential equations. Differential algebra was introduced by Joseph Ritt in 1950. Open problems The biggest open problems in the field include the Kolchin Catenary Conjecture, the Ritt Problem, and The Jacobi Bound Problem. All of these deal with the structure of differential ideals in differential rings. Differential ring A ''differential ring'' is a ring R equipped with one or more ''derivations'', whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
